Since Breaker's element has a highest value, the potential-sum always decreases. Therefore, if the initial potential-sum is less than 1, Breaker wins. Infinite boards The definition of Maker-Breaker game has a subtlety when there are infinitely many vertices () and infinitely many winning sets (). In this case we say that Breaker has a winning strategy if, for all j > 0, Breaker can prevent Maker from completely occupying a winning set by turn j. See also Clique game Arithmetic progression game Biased positional game Avoider-Enforcer game References Positional games